It doesn't even need to be (pseudo-) random. You can trace a spiral pattern out from the origin like so: 1 vertical up, turn right, 1 horizontal right, turn dow

It doesn't even need to be (pseudo-) random. You can trace a spiral pattern out from the origin like so: 1 vertical up, turn right, 1 horizontal right, turn down, 2 vertical down, turn left, 2 horizontal left, turn up, 3 vertical up, turn right, 3 horizontal right, turn down, etc. The resulting spiral pattern will never repeat.